§ 1. Direct construction
335
x / b X • On putting b = 1 + c with c > 0 and writing n for the integer part of
x, so that n :::: x < n + 1, we have
by the binomial formula, whence x/b x :::: 2(n + l)/n(n - 1)c 2 , an expression
which tends to 0 as n, i.e. x, increases indefinitely.
Finally, (9) reduces to (7) since, on putting loga x = y, we have x = a Y
and X S = a SY = bY with b = as, whence (loga x)/x s = y/b Y , which tends to 0
if b > 1 i.e. if a > 1; for a < 1 use loga x = -logl/a x.
The results are similar for the functions which tend to 0 when x -? +00:
(5.10)
(5.11)
(5.12)
o(b X ) when x -? +00 if 0 < a < b < 1;
if 0 < a < 1, s < 0;
if s < t < o.
Relation (10) is proved like (6). (11) is obtained by putting s = -t and
a = l/b, so that aX /x s = x- t /b x which tends to 0 by (7). Finally, (12) is
proved like (8).
We have not written the trivial relations between a function which tends
to 0 and a function which tends to +00.
What happens when x -? -00 can be deduced immediately from the
preceding, and concerns only the exponential functions since the others presuppose x > o. We put x = -y and apply the results for the case where
y -? +00. For example,
(5.13)
b X = o(a X ) when x -? -00 if 0 < a < b.
We have still to consider their behaviour on a neighbourhood of x = 0,
and this reduces to the behaviour at infinity on putting x = l/y. We find
(5.14)
(5.15)
when x -? +0 if s > t;
if s > o.
Some of these formulae may be written in terms of limits:
(5.16)
lim xSa- x = 0
if a > 1, s E JR,
x--->+oo
(5.17)
lim x- s loga x = 0
x--++(X)
if s > 0, a> 0,
(5.18)
lim X S loga x = 0
x--->O
if s > 0, a> o.
For example, (17) shows that (loga n)/n s tends to 0 to infinity for any s > 0,
and (18) shows that if you multiply log x, which tends to -00 as x tends to 0,
by a function which tends to 0 as slowly as Xl/lOO 000 000, the result tends to o.